The basic command is decompose.
For an additive model decompose(name of series, type = "additive").
For a multiplicative decomposition decompose(name of series, type ="multiplicative").

 Important first step: As a preliminary you have to use a ts command to define the seasonal span for a series.

For quarterly data, it might be name of series = ts(name of series, freq = 4).
For monthly data, it might be name of series = ts(name of series, freq = 12).

You can plot the elements of the decomposition by putting the decompose command as an argument of a plot command.

As an example,

plot(decompose(earnings, type = "multiplicative"))

Another way to plot is to store the results of the decomposition into a named object and then plot the object. As an example,

decompearn = decompose(earnings, type = "multiplicative")
plot(decompearn)

To see all elements of a stored object, simply type its name. For instance, entering decompearn will show all elements of the decomposition in the example above.

When the decomposition is stored in an object, you also have access to the various elements of the decomposition. For instance, in the example just given, decompearn\$figure contains the seasonal effects values for four quarters.

You could “print” the seasonal figures simply by entering decompearn\$figure. You could plot them using plot(decompearn\$figure).

beerprod = scan("beerprod.dat")
beerprod = ts(beerprod, freq = 4)
decompbeer = decompose (beerprod, type = "additive")
plot (decompbeer)
decompbeer

Using the Seasonal Values

The elements of \$figure are the effects for the four quarters.

The seasonal values are used to seasonally adjust future values. Suppose for example that the next quarter 4 seasonal value past the end of the series has the value 535. The quarter 4 seasonal effect is 57.433088, or about 57.43. Thus for this future value, the “de-seasonalized” or seasonally adjusted value = 535 − 57.43 = 477.57.

How the Trend Values Were Calculated

The trend values were determined as “centered“ moving averages of span 4 (because there are four quarters per year). Here’s how the centered moving average for time = 3 would be calculated.

Average the observed data values at times 1 to 4:

\(\dfrac{1}{4}(x_1+x_2+x_3+x_4)\)

Average the values at times 2 to 5:

\(\dfrac{1}{4}(x_2+x_3+x_4+x_5)\)

Then average those two averages:

\begin{multline} \dfrac{1}{2}\left(\dfrac{1}{4}(x_1+x_2+x_3+x_4)+\dfrac{1}{4}(x_2+x_3+x_4+x_5)\right) \\ \shoveleft{= \dfrac{1}{8}x_1+\dfrac{1}{4}x_2 + \dfrac{1}{4}x_3 +\dfrac{1}{4}x_4 + \dfrac{1}{8}x_5} \end{multline}

More generally, the centered moving average smoother for time t (with 4 quarters) is

\(\dfrac{1}{8}x_{t-2}+\dfrac{1}{4}x_{t-1} + \dfrac{1}{4}x_t +\dfrac{1}{4}x_{t+1} + \dfrac{1}{8}x_{t+2}\)

Following are the first 8 values in the observed series. The smoothed trend value for time 3 in the series (Qtr 3 of year 1) is 255.325 and the smoothed trend value for time 4 is 254.4125. Use the data below to verify these values (and your understanding of the procedure).

For monthly data the centered moving average smoother for time t will be

\(\dfrac{1}{24}x_{t-6}+\left(\sum_{j=-5}^{5}\dfrac{1}{12}x_{t+j}\right)+\dfrac{1}{24}x_{t+6}\)

decombeermult = decompose (beerprod, type = "multiplicative")
decombeermult$figure

Lowess Seasonal and Trend Decomposition

A lowess smoother essentially replaces values with a “locally weighted” robust regression estimate of the value. The R command stl does an additive decomposition in which a lowess smoother is used to estimate the trend and (potentially) the seasonal effects as well. There are several parameters that can be adjusted, but the default does a fairly good job.

For our beer production example, the following command works:

stl(beerprod, "periodic")

The “periodic” parameter essentially causes the seasonal effects to be estimated in the usual way, as averages of de-trended values. The alternative to this is a s.window = some odd number of lags, which uses lowess smoothing procedures to estimate the seasonal effects based on a number of years = the specified s.window value. When you do this, the seasonal effects will change as you move through the series.

Here’s a piece of the result of the stl(beerprod, "periodic") command.

The additive seasonal effects are 8.06289, -41.58529, -24.68456, 58.20698.

These aren’t much different than what we got from the additive decompose. Those seasonal values were

\$figure

(1) 7.896324 -40.678676 -24.650735 57.433088

The command plot(stl(beerprod, "periodic")) gave the following plot.